In order to set up a numerical approximation for (REF ), we consider a discrete grid of nodes \((x_{i},q)\)  in the state space \(\Omega \)  with discretization parameters \(\Delta x\)  that for simplicity we consider as the open set \([a,b]\times [c,d]\) , with \(a,b,c,d \in \operatorname{\mathbb {R}}\) . Thus, a general \(i=(i_1,i_2)\)  \(x_{i}=(a+i_1 \Delta x , c+i_2\Delta x)\) . We introduce an auxiliary parameter \(h\)  which has the role of tuning the distance of the search of information in the upwind direction and it should be taken, generally, of order \(O(\Delta x)\)  to maximize the accuracy of the approximation [1]}. In what follows, we denote the discretization steps in compact form by \(\Delta =(h, \Delta x)\)  and the approximate value function by \(V^\Delta \) .
